Revista de la
Unión Matemática Argentina
On the differential and Volterra-type integral operators on Fock-type spaces
Tesfa Mengestie
Volume 63, no. 2 (2022), pp. 379–395    

https://doi.org/10.33044/revuma.2149

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Abstract

The differential operator fails to admit some basic structures including continuity when it acts on the classical Fock spaces or weighted Fock spaces, where the weight functions grow faster than the classical Gaussian weight function. The same conclusion also holds in some weighted Fock spaces including the Fock–Sobolev spaces, where the weight functions grow more slowly than the Gaussian function. We consider modulating the classical weight function and identify Fock-type spaces where the operator admits the basic structures. We also describe some properties of Volterra-type integral operators on these spaces using the notions of order and type of entire functions. The modulation operation supplies richer structures for both the differential and integral operators in contrast to the classical setting.

References

  1. A. Aleman and J. A. Cima, An integral operator on $H^p$ and Hardy's inequality, J. Anal. Math. 85 (2001), 157–176. MR 1869606.
  2. A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. MR 1481594.
  3. M. J. Beltrán, Dynamics of differentiation and integration operators on weighted spaces of entire functions, Studia Math. 221 (2014), no. 1, 35–60. MR 3194061.
  4. J. Bonet, Dynamics of the differentiation operator on weighted spaces of entire functions, Math. Z. 261 (2009), no. 3, 649–657. MR 2471093.
  5. J. Bonet and A. Bonilla, Chaos of the differentiation operator on weighted Banach spaces of entire functions, Complex Anal. Oper. Theory 7 (2013), no. 1, 33–42. MR 3010787.
  6. J. Bonet, T. Mengestie and M. Worku, Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces, Results Math. 74 (2019), no. 4, Paper No. 197, 15 pp. MR 4040629.
  7. O. Constantin, A Volterra-type integration operator on Fock spaces, Proc. Amer. Math. Soc. 140 (2012), no. 12, 4247–4257. MR 2957216.
  8. O. Constantin and J. Peláez, Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces, J. Geom. Anal. 26 (2016), no. 2, 1109–1154. MR 3472830.
  9. D. Girela and J. Peláez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006), no. 1, 334–358. MR 2264253.
  10. A. Harutyunyan and W. Lusky, On the boundedness of the differentiation operator between weighted spaces of holomorphic functions, Studia Math. 184 (2008), no. 3, 233–247. MR 2369141.
  11. S. Janson, J. Peetre and R. Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamericana 3 (1987), no. 1, 61–138. MR 1008445.
  12. D. H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine's inequality, Michigan Math. J. 40 (1993), no. 2, 333–358. MR 1226835.
  13. T. Mengestie, A note on the differential operator on generalized Fock spaces, J. Math. Anal. Appl. 458 (2018), no. 2, 937–948. MR 3724708.
  14. T. Mengestie, Carleson type measures for Fock–Sobolev spaces, Complex Anal. Oper. Theory 8 (2014), no. 6, 1225–1256. MR 3233976.
  15. T. Mengestie, On the spectrum of Volterra-type integral operators on Fock–Sobolev spaces, Complex Anal. Oper. Theory 11 (2017), no. 6, 1451–1461. MR 3674286.
  16. T. Mengestie, Spectral properties of Volterra-type integral operators on Fock–Sobolev spaces, J. Korean Math. Soc. 54 (2017), no. 6, 1801–1816. MR 3718425.
  17. T. Mengestie, Volterra type and weighted composition operators on weighted Fock spaces, Integral Equ. Oper. Theory 76 (2013), no. 1, 81–94. MR 3041722.
  18. T. Mengestie and S.-I. Ueki, Integral, differential and multiplication operators on generalized Fock spaces, Complex Anal. Oper. Theory 13 (2019), no. 3, 935–958. MR 3940399.
  19. T. Mengestie and M. Worku, Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces, Integral Transforms Spec. Funct. 30 (2019), no. 1, 41–54. MR 3871993.
  20. J. Pau and J. Peláez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights, J. Funct. Anal. 259 (2010), no. 10, 2727–2756. MR 2679024.
  21. S.-I. Ueki, Characterization for Fock-type space via higher order derivatives and its application, Complex Anal. Oper. Theory 8 (2014), no. 7, 1475–1486. MR 3261707.
  22. R. Wallstén, The $S^p$-criterion for Hankel forms on the Fock space, $0<p<1$, Math. Scand. 64 (1989), no. 1, 123–132. MR 1036432.
  23. K. Zhu, Analysis on Fock Spaces, Graduate Texts in Mathematics, 263, Springer, New York, 2012. MR 2934601.